This page is here just to provide a pointer to the page for the 2015-16 course.  (You can also click on the tab above.)


The September 2015 version is in the usual place.  I’m posting it because our academic year is just starting, and I will be teaching Math 216 again; this year’s course website is here.

The list of intended changes and corrections has grown again, but essentially all are small.  My intent is to try to have the changes and corrections in each chapter digested as much as possible before the courses reaches there.

The one bit of potentially new content:  David Speyer pointed out that Grobner bases are something that people could and should reasonably see in a first course.  Over lunch in Utah, I thought it through with him and Kiran Kedlaya and Tom Graber.  I will likely post a draft here before thinking about whether to including it.

My hope of a new version each month has clearly not worked out! But here is one for April, just before the month ends. As usual, a random selection of corrections and minor improvements have been dealt with, and many more remain to do.

The April version is posted at the usual place (the April 29, 2015 version).

I am going to attempt to put up a version every month, in order to ensure that I continue to spend a little time tidying things up every so often.  So the January version is now  posted at the usual place (the January 29, 2015 version, also in e-reader format).

The list of things to do has shrunk, but there is a good deal still to do.  I have had many useful detailed comments to later digest and then include from Gurbir Dhillon, Tony Feng, Lisa Sauermann, and Jesse Silliman; and Benjamin Ljundberg, Chandrasekhar Raju, and Scott Zhang.


The last version posted was in 2013. I wanted to get the next version out in 2014, and time is running out!  So here it is, posted at the usual place (the Dec. 30, 2014 version).  (There is also a version suited for an e-reader — thanks to Jack Sherk asking for it, and explaining to me how to make it.)

In this 2014 post, I can’t help mentioning the passing of Grothendieck.   I was more moved than I expected to hear the news and feel the ripples in the mathematical community.  It feels strange that he is now a historical figure, even though he had walked away from his unfinished cathedral long before I was even aware of its existence.

The notes have continued to evolve around the edges, although the material is stable. Please continue to give me corrections and suggestions!  There are few sections that need tender loving care; I mention them on the front.  There are many other things on my to-do list as well, including many comments you’ve made.

The notes now have a tentative title (The Rising Sea:  Foundations of Algebraic Geometry).  The phrase is due to Grothendieck, translated by Colin McLarty; it is the title of Daniel Murfet’s wonderful blog.

The index is in the process of being made.  (No need to give me specific comments until it has converged!)

Finally, if you would like notes on commutative algebra that are very much from the same point of view as these notes, you may enjoy Andy McLennan’s notes, available here.  It also contains an (explicated) English translation of Serre’s epochal FAC, and all the algebra needed as background.

The course webpage is here. The course notes are here.

A new version is now posted at the usual place (the June 11, 2013 version).   The main reason for this post is to have a reasonably current version on the web.  Thanks to many people for helpful comments — most recently, a number of comments from János Kollár, Jeremy Booher, Shotaro (Macky) Makisumi, Zeyu Guo, Shishir Agrawal, Bjorn Poonen, and Brian Lawrence (as well as many people posting here, whose names I thus needn’t list).

There are a large number of very small improvements, and I’ll list only a few in detail.  I’ve replaced the proof of the Fundamental Theorem of Elimination Theory (Theorem 7.4.7).  (This new proof is much more memorable for me.  It’s also shorter and faster, and generally better.  It was the proof I was trying to remember, which I heard in graduate school.  I couldn’t find it in any of the standard sources, and reproduced it from memory.  But it must be in a standard source, because it is certainly not original with me!)   Jeremy Booher’s comments have led to the completion discussion being improved a lot.

I had earlier called \rm{Spec} A, where A is the subring k[x^3, x^2, xy, y] of k[x,y], the “knotted plane”, to suggest the picture that it was a plane, with the origin somehow “knotted” or “pinched”.  János Kollár pointed out that “knotted” is misleading, because it is not in any obvious way knotted.  I’ve changed this to “crumpled plane”, but this isn’t great either.  I’m now thinking about “pinched plane”.  Does anyone have a good suggestion for a name for this important example?

Still to do:  To repeat my comments from the previous post, as usual, the figures, index, and formatting have not yet been thought about.  My to-do list is quite short, so please complain about anything and everything (except figures, index, and formatting).  (Perhaps in the next posted version, I may have “TODO” appearing in the text indicating where there is still work in progress.)

Finally, just for fun, here is a picture of the two rulings on the quadric surface, from the Kobe skyline.  (It is clear why algebraic geometry is so strong in Japan!)

Kobe port

Kobe port (click to enlarge)